Magnetic bottles for the Neumann problem : curvature effects in the case of dimension <span class="mathjax-formula">$3$</span> (general case)

4^esérie, t. 37, 2004, p. 105 à 170.

MAGNETIC BOTTLES FOR THE NEUMANN PROBLEM:

CURVATURE EFFECTS IN THE CASE OF DIMENSION 3 (GENERAL CASE)

B

B

HELFFER

A

MORAME

ABSTRACT. – In a recent paper Lu and Pan have analyzed the asymptotic behavior, in the semi-classical regime, of the ground state energy of the Neumann realization of the Schrödinger operator in the case of dimension 3. Although these results are rather satisfactory when the magnetic field is non-constant and satisfies some generic conditions, they are not sufficient in the case of a constant magnetic field for understanding phenomena like the onset of superconductivity and more accurate results should be obtained.

In the two-dimensional case, the effects due to the curvature of the boundary were predicted by a formal analysis of Bernoff–Sternberg and finally proved by the joint efforts of Lu–Pan, Del Pino–Felmer–Sternberg and Helffer–Morame. Our aim is to analyze similar effects in dimension3. As known from physicists and roughly analyzed by Lu–Pan, it turns out that the results depend on the geometry of the boundary especially at the points where the magnetic field is tangent at the boundary. We present here the analog of the Bernoff–

Sternberg conjecture (also formulated in a different form by Pan) and prove it in the generic situation.

2004 Published by Elsevier SAS

RÉSUMÉ. – Récemment Lu et Pan ont analysé le comportement asymptotique, en régime semi-classique, de la plus petite valeur propre de la réalisation de Neumann d’un opérateur de Schrödinger magnétique dans le cas de la dimension3. Si les résultats obtenus sont satisfaisants lorsque le champ magnétique est variable et vérifie quelques conditions génériques, ils ne permettent pas, dans le cas d’un champ magnétique constant, de comprendre des phénomènes comme la localisation d’une fonction propre associée. Dans le cas de la dimension2, Bernoff et Sternberg avaient conjecturé, sur la base de constructions formelles, que c’était la courbure du bord qui allait être la clef de cette localisation. Ceci fut finalement prouvé par Lu–Pan, Del Pino–Sternberg et Helffer–Morame. Nous nous proposons ici d’analyser le même problème en dimension3.

La littérature physique indique (et les premiers résultats dans cette direction furent démontrés par Lu et Pan) que tout dépend de la géométrie de la frontière et plus précisément de la courbe (génériquement) où le champ magnétique est tangent au bord. Nous présentons ici l’analogue de la conjecture de Bernoff–

Sternberg dans le cas de la dimension3(conjecture aussi formulée sous une forme un peu différente par Pan) et en donnons la démonstration sous des hypothèses génériques.

2004 Published by Elsevier SAS

1This work has been partially supported by the European Union under the TMR Program EBRFXT 960001 and the ESF program SPECT.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

1. Introduction

Our study is motivated by a question in superconductivity.² Let us roughly explain how the question occurs in this context (see [10] or [22] for a review on this aspect and references therein).

Given a bounded open setΩ⊂R³with smooth boundary, we are interested in the properties of the minimizers of the so called Ginzburg–Landau functional

H¹(Ω;C)×H¹(Ω;R³)(ψ,A) → G(ψ,A) with:

G(ψ,A) =

Ω

(∇ −iκA)ψ ²+κ² 2

|ψ|²−12 dx+κ²

Ω

|curlA−σcurlA|²dx.

(1.1)

HereAis a given applied magnetic potential inH¹(Ω;R³),σis a positive parameter introduced for permitting a variation of its intensity and the parameterκ >0 reflects the properties of the sample Ωwhich will be assumed to be large. AsΩis bounded, the existence of a minimizer is rather standard. It is also easy to observe that the pair (0, σA)is a trivial critical point of the functionalGwhich is called a normal solution. It has also be shown, that this pair is, when curlA:=H does not vanish, a global minimum forσ large enough. It is therefore natural to discuss in function ofσ, if this pair is a local or a global minimizer. By looking at the Hessian of the functional at this point, this question is immediately related to the positivity of the operator:

−(∇ −iκσA)²−κ² inΩ,

where we get from the minimization of the functional a “magnetic” Neumann condition at the boundary.

It can be seen that the change of sign of the lowest eigenvalue of this operator occurs forσ of the order of κ. To get a more precise information, we are immediately let to analyze the groundstate energy (that is the lowest eigenvalue) of the Neumann realization of the Schrödinger operator with magnetic potential in an open setΩinR³:

P_A^h= 3 j=1

hDx_j −Aj(x)2

, (1.2)

whereh=_κσ¹ will be a small parameter. This is this problem which will be the main object of the paper. We recall that an elementuinH²(Ω)satisfies the (magnetic) Neumann condition at the boundary, if

N(x)·

(hD−A)u

(x) = 0, ∀x∈∂Ω, whereN(x)is the normal atxto∂Ω.

2One can find in Chapter 4 of [30] a physical presentation of the problem we are considering. We particularly emphasize on their Section 4.3 where they analyze (with partially heuristic arguments) the angular dependence of the nucleation field. We will give a mathematical proof of, what they describe for example p. 87: “For type II superconductors (The authors meanκ1/√

2, but we treat onlyκlarge) the above calculation shows that superconductivity is not entirely destroyed forHc₂< H < Hc₃. A superconducting sheath remains close to the surface parallel to the applied field. Conversely, when the field is decreased below Hc₃, a superconducting sheath appears at the surface before superconductivity is restored in the bulk atH=Hc2. If the sample is a long cylinder with the applied field parallel to the axis, the sheath will cover all the surface of the cylinder. If it is a sphere, the sheath will be restricted to a small zone near the equatorial plane whenHis close toHc₃. When the field is decreased towardsHc₂the sheath will progressively extend up to the poles”.

This realization will be denoted by P_A,Ω^h,N but we will sometimes use shorter notations when there is no ambiguity. If λ(h) is the lowest eigenvalue and u^h(x) is a corresponding normalized eigenstate, we would like to discuss the asymptotic of λ(h) as h→0, with the hope to analyze in a future work the localization of u^h. This problem was treated in the case of dimension2by Bernoff–Sternberg [3], Lu–Pan [19–21], Del Pino–Felmer–Sternberg [28] and Helffer–Morame [12] in a rather complete way. Later a recent work by Lu–Pan [23] was starting the analysis of the dimension3and this was complemented slightly in [13]. So this paper is the continuation of the program, with a particular emphasis on the constant magnetic field case.

We start by establishing rough estimates, but sufficiently accurate for determining the effective order of the curvature effect.

THEOREM 1.1. – LetΩbe a bounded open set ofR³withC^∞boundary∂Ω. LetP_A,Ω^h,N be the Neumann operator onL²(Ω)associated to the Schrödinger operator with constant magnetic field(hD−A)²and let us assume that the vector magnetic fieldH= curl(A)is constant and of intensity equal tob=|H|.

If

λ(h) = inf Sp(P_A,Ω^h,N)

is the first eigenvalue ofP_A,Ω^h,N, then there exists a constantC0such that λ(h)−bΘ0hC0h^4/3,

(1.3)

whereΘ0∈]0,1[is an universal constant, which will be defined in (2.4).

We observe that in the case of dimension2, the corresponding error was inO(h^3/2).

We then continue with the analysis of the curvature effects, which are the analog in dimension 3 of the results first conjectured by Bernoff–Sternberg (see [3]) in the case of dimension2. Before establishing the corresponding conjecture in the case of dimension3, let us describe the main ingredients appearing in the assumptions.

It has been observed by [23] and this will be recalled in Section 4, that the ground stateu^his localized (ash→0) near the boundary∂Ωbut more precisely on the set:

ΓH= x∈∂Ω|

H|N(x)

= 0 , (1.4)

that is the set of points in∂ΩwhereH is tangent.

It is natural to assume that:

ΓHis a regular submanifold of∂Ω, (1.5)

that is a disjoint union of regular curves. From now on, we choose an orientation on each curve. At each pointxofΓH, we will associate the normal curvature along the magnetic field H=H(B)by:

κn,B(x) :=Kx

T(x)∧N(x), H

|H|

, (1.6)

whereKdenotes the second fundamental form on the surface∂Ω(see Section 8) andT(x)is the unit oriented tangent vector toΓH atx. It is natural to assume that:

κn,B= 0, onΓH. (1.7)

The last generic assumption is:

The set of points whereHis tangent toΓHis isolated.

(1.8)

The following function onΓH will play an important role:

˜ γ0(x) =

1 2

2/3

ˆ

ν0δ₀^1/3κn,B(x)^2/3

δ0+ (1−δ0)

T(x) H

|H|

²1/3 , (1.9)

whereνˆ0>0 andδ0∈]0,1[ are universal constants attached to spectral invariants related to two model Hamiltonians respectively defined onRandR⁺and which will be defined in (2.15) and (2.7).

Associated to this function, which will play the role of effective curvature, we define:

ˆ γ0= inf

x∈ΓH

˜ γ0(x).

(1.10)

Our main theorem is:

THEOREM 1.2. – LetP_A,Ω^h,Nbe the Neumann realization onL²(Ω)of the magnetic Laplacian (hD−A)², whereh∈]0,1[is a small parameter andAcorresponds to constant magnetic field.

Under assumptions (1.5), (1.7) and (1.8), there existη >0andγˆ0>0such that:

inf Sp(P_A,Ω^h,N) =bΘ0h+ ˆγ0b^2/3h^4/3+O(h^43+η), (1.11)

whereγˆ0is defined in (1.10).

We conjectured this theorem in September 2001, simultaneously with Pan [26] (who proposed another equivalent³ formulation and obtained the upper bound). A first complete proof was given in mp_arc [14] under additional non generic conditions. This paper, which is a slightly modified version of the preprint [15], gives now a complete proof in the general generic case. Although the methods of proof can also lead to localization results for the ground state (see [12,13]) or more generally for minimizers of the Ginzburg–Landau functional (see [19–23,17]), this will not be discussed here. This was actually explored in [26], under the assumption that the conjecture was true.

The paper is organized as follows.

In Sections 2 and 3, we recall previous results extracted from [6,12,23,13], which will play an important role in the analysis. In Section 4, we recall the results of [23] devoted to the case when the magnetic field is not constant. It is also shown there that the problem is reduced in the constant magnetic case to a neighborhood of the boundary. In Section 5, we make explicit our choice of coordinates near the boundary. Section 6 gives rough upper bounds for the ground state energy by constructing quasimodes. In Section 7, we present our first lower bounds which are sufficiently accurate for giving the right order for the remainder. In Section 8, we go further in the choice of adapted coordinates, taking in particular account of the fact that the magnetic field is constant. This permits to introduce our magnetic invariants attached toΓHand to present our main results in a more precise form. Section 9 is devoted to the research of simpler models for the magnetic potentials obtained in the adapted coordinates by suitable gauge transformations and by neglecting “small” terms. Section 10 is devoted to the estimate of the errors done in the

3See the appendix in [15].

use of the previous models. In Section 11, we start by some heuristics which lead to the spectral analysis of a simplified model which will be used in the general case. Section 12 is devoted to the proof of the accurate upper bounds. Section 13 gives complementary lower bounds for the model. Section 14 presents the structure of the proof of accurate lower bounds according to different zones and treats the easy zones. Sections 15 and 16 correspond to the treatment of the difficult zones.

2. The case of dimension2 2.1. Main results for the models with constant magnetic fields

We first consider the operator:

P_A^h:= (hDx1−A1)²+ (hDx2−A2)², with

A=

−b 2x2,b

2x1

, h >0 and b >0,

and analyze the spectrum of its realization inR²or of its Neumann realization inR²+.

We observe that by homogeneity, one can reduce the analysis toh= 1andb= 1. It is well known that in the case of R² the spectrum is a point spectrum and that the eigenvalues are given by(2n+ 1)with (n∈N), each eigenvalue being of infinite multiplicity. One way, to see this is to show the unitary equivalence (via a gauge transformation, a partial Fourier transform and a translation) with the harmonic oscillator (D_t²+t²)but seen as an unbounded operator onL²(R²t,s).

The case of the Neumann realization inR²+is a little more delicate. By unitary transformation we get the Neumann realization of the operator of

Q(t, Dt;s) :=D_t²+ (t−s)² onL²(R+×R), (2.1)

which is reduced to the analysis of the family (s∈R) of operators:

Q(s) :=D²_t+ (t−s)², (2.2)

defined onL²(R+).

Letµ(s)be defined by:

µ(s)is the lowest eigenvalue of the Neumann realization ofQ(s)inR⁺. (2.3)

One easily shows (cf. [29])

PROPOSITION 2.1. – The infimum of the spectrum of the Neumann realization ofP_A^his given bybhΘ0with

Θ0:= inf

s∈Rµ(s).

(2.4)

2.2. An important one-dimensional family of operators onR

We recall the main properties of the groundstate energy of the Neumann realization onR⁺of Q(s) :=D²_t+ (t−s)²,

(2.5)

in function of the parameter s∈R. Let us just list some properties of µ(s) and of the corresponding normalized eigenvectorϕ^sand refer to [6,12] for proofs or details. The Neumann eigenvalueµ(s)satisfies the following properties:

1.lims→−∞µ(s) = +∞; 2.µis decreasing fors <0;

3.µ(0) = 1;

4.lims→+∞µ(s) = 1;

5.µadmits in]0,+∞[a unique minimumΘ0<1at someξ0>0. So Θ0:= inf

s∈Rµ(s) =µ(ξ0)<1.

(2.6)

This minimum is nondegenerate and more precisely:

0< δ0:=1

2µ(ξ0)<1.

(2.7)

6. We introduceϕ0=ϕ^ξ0. We have:

R+

(t−ξ0)ϕ0(t)²dt= 0, (2.8)

which just corresponds to the conditionµ(ξ0) = 0.

7.ϕ0is rapidly decreasing at∞.

Let us just mention that the proof of some of these properties is based on the following identity, relating the first eigenfunctionϕ^sandµ(s):

ϕ^s2µ(s) =

s²−µ(s) ϕ^s(0)2

. (2.9)

2.3. Applications: main results in dimension2

As an application of the analysis of the models, one gets (cf. [21]) via a partition of unity for the lower bound and a suitable construction of quasimodes for the upper bound, the following general result (in the two-dimensional case):

THEOREM 2.2. – Ifλ(h) is the lowest eigenvalue of the Neumann realization ofP_A^hinΩ, then we have:

hlim→0

λ(h)

h = min

xinf∈Ω

B(x),Θ0 inf

x∈∂Ω

B(x), (2.10)

whereB= curlA.

We emphasize that there is no assumption that the magnetic field is constant. In the case of the constant magnetic field, one can actually have more precise results forλ(h). WhenBis constant, the minimum in (2.10) is obtained by the second term and we showed in [3,28,12] the

THEOREM 2.3. – If the magnetic field is constant of intensityb, then:

λ(h) =bΘ0h−c1b^1/2 sup

x∈∂Ω

κ(x)

h^3/2+ o(h^3/2), (2.11)

wherec1is a universal strictly positive universal constant and κ(x)is the curvature of∂Ωat x∈∂Ω.

2.4. The Montgomery’s model and a second important family of operators onR

We just discuss a model that we shall meet indirectly later and which is interesting. It first appears in [24] but see also [11]. We consider inR²x,y, and for some parameterκ >0the operator:

P:=h²D²_x+

hDy−κ 2x²

2

. (2.12)

The magnetic potential isA&= (0,^κ₂x²)and we have:

curlA=κx.

So the magnetic field vanishes along the line {x= 0}. Let us briefly describe the spectral analysis. After a Fourier transform in they-variable, we first get:

Pˆ=h²D_x²+

hη−κ 2x²

2

,

and this leads to the analysis of the family, parametrized by η∈R, of selfadjoint operators onL²(R):

Pˆ(η) =h²D²_x+

hη−κ 2x²

2

. Using a simple dilation, we get:

inf Sp( ˆP) = inf

η inf SpPˆ(η)

=h^4/3 κ

2 ^2/3inf

ρ inf Sp

D²_r+ (r²−ρ)² . (2.13)

Let us recall some properties of the family of operators S(ρ) =D²r+ (r²−ρ)², (2.14)

and of the corresponding ground stateψ^ρ, which were established in [24,11] and [27].

1. There exists a uniqueρ=ρminsuch that:

ˆ ν0:= inf

ρ inf Sp

D_r²+ (r²−ρ)²

= inf Sp

D²_r+ (r²−ρmin)² . (2.15)

2.ψ^ρbelongs toS(R)and is even.

We shall later use the notation:

ψ0=ψ^ρmin. (2.16)

3. Constant magnetic field: models inR³and inR³+

As in the case of dimension2, where the first thing was to understand the model with constant magnetic field inR²andR²+, we shall discuss the case ofR³andR³+.

3.1. Model inR³

We start (after some gauge transformation) of the Schrödinger operator with constant magnetic field in dimension3.

P^h(H) :=& h²D²_x1+ (hDx₂−H3x1)²+ (hDx₃+H2x1−H1x2)². (3.1)

The following lemma has the age of quantum mechanics:

LEMMA 3.1. – The bottom of the spectrum of the selfadjoint realization ofP^h(H&)inR³is inf Sp

P_R^h,N3 (H&)

=bh, (3.2)

where

b=|H|=

H₁²+H₂²+H₃² is the intensity ofH.

3.2. Models in halfspaces

We refer to [23] and [13] for the proof of the results presented in this subsection. IfNis a unit vector inR³, we now consider the Neumann realization inΩ :={x∈R³|x·N >0}. After a rotation, we can assume in the proofs thatN= (1,0,0), soΩisR³+:={x1>0}.

After scaling, we can assume thath= 1and|H|= 1.

After some rotation in the (x2, x3)variables, we can assume that the new magnetic field is (β1, β2,0)and we are reduced to the problem of analyzing:

P(β1, β2) :=D_x²₁+D²_x2+ (Dx₃+β2x1−β1x2)², in{x1>0}, where:

β₁²+β²₂= 1.

We introduce:

β2= cosϑ, β1= sinϑ, and we observe that, ifNis the external normal tox1= 0, we have:

H& |N=−sinϑ.

(3.3)

By partial Fourier transform, we arrive to:

L(ϑ, τ) =D²_x1+D²_x2+ (τ+ cosϑx1−sinϑx2)², (3.4)

inx1>0 and with Neumann condition onx1= 0. The bottom of the spectrum of L(ϑ, τ)is given by:

σ(ϑ) := inf Sp

L(ϑ, Dt)

= inf

τ

inf Sp

L(ϑ, τ) . (3.5)

PROPOSITION 3.2. – The bottom of the spectrum of the Neumann realization ofP^h(H&)in Ω :={x∈R³|x·N >0}is:

inf SpP_Ω^h,N(H&) =σ(ϑ)bh, (3.6)

whereϑ∈[−^π₂,^π₂]is defined by (3.3).

By symmetry considerations, we observe also that:

σ(ϑ) =σ(−ϑ) =σ(π−ϑ).

(3.7)

It is consequently enough to look at the restriction to[0,^π₂].

3.3. Properties ofϑ→σ(ϑ)

Let us now list the main properties of the function ϑ→σ(ϑ) on [0,^π₂]. Most of them are established in [23] but see also [13].

1.σis continuous on[0,^π₂].

2.

σ(0) = Θ0<1.

(3.8) 3.

σ π

2

= 1.

(3.9) 4.

σ(ϑ)Θ0(cosϑ)²+ (sinϑ)². (3.10)

5. If ϑ∈]0,^π₂[, the spectrum ofL(ϑ, τ) is independent of τ and its essential spectrum is contained in[1,+∞[.

6. Forϑ∈]0,^π₂[,σ(ϑ)is an isolated eigenvalue ofL(ϑ, τ), with multiplicity one.

7. The functionσis strictly increasing on[0,^π₂[.

8. The functionσhas the following expansion forϑsmall:

σ(ϑ)∼Θ0+

n1

αn|ϑ|ⁿ, (3.11)

withα1=√ δ0=

µ(ξ0)/2>0.

A first consequence of this analysis is

PROPOSITION 3.3. – When b=|H| is fixed the bottom of the spectrum of P_Ω^h,N(H&) in Ω :={x·N >0}is minimal whenϑ= 0that is, according to (3.3), that is when the magnetic field vector satisfiesH& ·N= 0.

4. First results for general magnetic fields

In the case of dimension2, under the assumption thatcurlA:=B(x)>0, the basic estimate at the interior was the inequality:

h

B(x)u(x)²dx (h∇ −iA)u²dx, ∀u∈C₀^∞(Ω).

(4.1)

This was not enough for understanding the Neumann problem. One should more carefully analyze the case ofR²+ and, in the case of the constant magnetic field, one should also analyze more complicate models (like for example the case of the disk). The most spectacular result was:

THEOREM 4.1. – When the magnetic field is constant, the groundstate of the Neumann realization ofP_A^h in an open regular bounded setΩ⊂R² is localized in the neighborhood of the points of the boundary of maximal curvature.

In the case of dimension 3, estimate (4.1) should be replaced by the weaker estimate established in [11] (Theorem 3.1)

THEOREM 4.2. – There existCandh0>0such that, for allh∈]0, h0], we have:

h

Ω

H(x)−Ch^1/4u(x)²dx

Ω

(h∇ −iA)u²dx, ∀u∈C0^∞(Ω).

(4.2)

If this result is essentially sufficient for analyzing the Dirichlet problem inΩ, it is necessary to implement the analysis given in the first part in order to treat the Neumann problem. Near each point of the boundaryx, we have to use the lower bound obtained for the model with constant magnetic fieldH=H(x). Following for example the proof in [12] and using the same partition of unity, we get:

THEOREM 4.3. – h

Ω

Wh(x)u(x)²dx (h∇ −iA)u²dx, ∀u∈H¹(Ω), (4.3)

where

Wh(x) =

|H(x)| −Ch^1/4, if d(x, ∂Ω)2h^3/8,

|H(s(x))|σ(ϑ(x))−Ch^1/4 if d(x, ∂Ω)2h^3/8. (4.4)

Here we recall thatϑ(x)satisfies:

H

s(x)·sinϑ(x) =− H

s(x)

|N s(x)

, (4.5)

wheres(x)is, forxnear∂Ω, the point in∂Ωsuch that:

d(x, ∂Ω) =d x, s(x)

, and we observe that, due to (3.7),σ(ϑ(x))is well defined by (4.5).

The first consequence (compare with Theorem 2.2) is:

THEOREM 4.4. –

hlim→0

λ(h)/h

= min

xinf∈Ω

H(x), inf

x∈∂Ω

H(x)σ ϑ(x)

. (4.6)

The lower bound is a direct consequence of Theorem 4.3 and the proof of the upper bound is sketched in [23]. WhenH(x)is constant of intensityb, then the minimum in (4.6) is obtained for the second term and we have:

hlim→0

λ(h)/h

=b

xinf∈∂Ωσ ϑ(x)

. (4.7)

In view of (6.3), there exists a pointxsuch thatϑ(x) = 0. So we get:

hlim→0

λ(h)/h

= Θ0b.

(4.8)

Remark 4.5. – As in [12], this leads to localization theorems ([23,13], see also [28]). In particular, if the magnetic field is constant, then the ground state is localized near the points of the boundary where the magnetic field is parallel to the tangent space.

Application. An interesting case is the case whenHis constant and whenΩis the ellipsoid:

a1x²1+a2x²2+a3x²31.

The set of points whereH& = (H1, H2, H3)is parallel is obtained by intersecting the boundary of the ellipsoid with the plane:

a1x1H1+a2x2H2+a3x3H3= 0.

More generally, if the surface is strictly convex and if the magnetic field is constant, it is possible to show that the set of points of the boundary whereH is parallel to the tangent space is aC^∞curve.

We emphasize that Theorem 4.4 does not explain all the situation. In the case with constant magnetic field it would be nice to show the role of some curvature in the localization as in the case of dimension2. This is actually our goal to give an answer to this question.

5. Adapted coordinates 5.1. Magnetic geometrical invariants

The standard coordinates on R³ will be denoted by x= (x1, x2, x3) and the standard flat metric byg0. We will also use.|.forg0(. , .)or forg₀(. , .), and|X|for(X|X)^1/2, (ifXis a vector field or a one-form). The standard volume onR³will be denoted by

ω3=dx³=dx1∧dx2∧dx3

and will fix also the orientation ofR³.

LetAbe a smooth magnetic potential one-form onΩ:

A= 3 j=1

Aj(x)dxj, Aj∈C^∞( Ω;R).

The magnetic fieldBis the two-form:

B=dA=

1i<j3

Bij(x)dxi∧dxj, Bij(x) =∂Aj

∂xi

(x)−∂Ai

∂xj

(x).

(5.1)

If neededBij(x)is extended as an antisymmetric matrixBx. The intensity of the magnetic field is the non-negative function

|B|(x) = sup |Bx(X, Y)|; X, Y ∈TxΩ, |X|=|Y|= 1

, ∀x∈Ω.

(5.2)

One associates to the magnetic fieldB, a vector fieldH& :=H(B)∈TΩby Bx(X, Y) =ω3

X, Y, Hx(B)

, ∀X, Y ∈TxΩ.

(5.3)

In the initial coordinates, we have:

H(B) = 3

j=1

bj(x) ∂

∂xj

, withb1=B23, b2=B31andb3=B12. (5.4)

Then the magnetic intensity satisfies:

|B|=H(B). (5.5)

We associate to the magnetic potentialAthe sesquilinear form onH¹(Ω):

u→q_A^h(u) =

Ω

√

−1h du+uA²dx³. (5.6)

Hereh∈]0,+∞[is a parameter. The associated differential operator is the magnetic Laplacian which is given in the standard coordinates by

P_A^h= (hD−A)²= 3

j=1

hDx_j−Aj(x)2

. (5.7)

5.2. Local coordinates

Ify= (y1, y2, y3)are new local coordinates, that is, if Vx0 is a neighborhood of some point x0∈Ωand ifx→y= Φ(x)is a diffeomorphism ofVx0 ontoO ⊂R³, then in the new basis (_∂y^∂

1,_∂y^∂

2,_∂y^∂

3)ofTVx₀, the standard metricg0and the volumeω3become g0=

1i,j3

gijdyi⊗dyj, (5.8)

and

ω3=|g|^1/2dy1∧dy2∧dy3, (5.9)

with

gij= ∂x

∂yi

∂x

∂yj

, X|Y=

1i,j3

gijXiYj, |g|= de t(gij), where

X=

j

Xj

∂

∂yj

and Y =

j

Yj

∂

∂yj

. The magnetic potential is given in the new coordinates by

A= 3 j=1

Ajdyj withAj= 3 k=1

Ak

∂xk

∂yj

. (5.10)

The magnetic field is also given by

B=

1i<j3

Bijdyi∧dyj withBij=∂Aj

∂yi −∂Ai

∂yj

. (5.11)

So the vector magnetic field becomes

H(B) = 3

j=1

˜bj

∂

∂yj

, (5.12)

with

˜b1=|g|^−1/2B23, ˜b2=|g|^−1/2B31, ˜b3=|g|^−1/2B12, (5.13)

and the intensity of the magnetic field becomes:

|B|=

i,j

gij˜bi˜bj

1/2

. (5.14)

The sesquilinear form takes the following form, forusupported inVx0, q_A^h(u) =

Vx0

|g|^1/2

1i,j3

g^ij[hDy_iu−Aiu]×[hDy_ju−Aju]dy³, (5.15)

and the associated differential operator is P_A^h=|g|^−1/2

1i,j3

(hDyj −Aj)· |g|^1/2g^ij(hDyi−Ai).

(5.16)

Hereg^ijis the inverse matrix of the matrixgij. 5.3. Adapted local coordinates near the boundary

Let φ(x) = (y1, y2)be local coordinates on the boundary andG the induced metric byg0

on ∂Ω in these coordinates. Then for ε >0 small enough (and modifying a little Vx0 if necessary), we can define local coordinates onVx0,

Φ :Vx₀→ S ×]0, ε[, Φ(x) = (y1, y2, y3), whereSis an open set inR², such that

y3(x) = dist

x, φ⁻¹(y1, y2)

= dist(x, ∂Ω);

(5.17) so

x=φ⁻¹(y1, y2) +y3N

φ⁻¹(y1, y2)

, ∀x∈ Vx0, (5.18)

whereN(x)is the interior unit normal to∂Ωat the pointx∈∂Ω.

Then we get by simple computation the form of the standard flat metric g0 in these new coordinates,

g0=

1i,j3

gijdyi⊗dyj

=dy3⊗dy3+G+ 2y3

1i,j2

∂N

∂yi

∂x

∂yj

dyi⊗dyj

+y₃²

1i,j2

∂N

∂yi

∂N

∂yj

dyi⊗dyj. (5.19)

In this case we have, using (5.10) and (5.18), A3=

3 j=1

AjNj.

Remark 5.1. – In the following, we choose a finite family of open setsVx_j(xj∈∂Ω) covering a tubular neighborhood of∂Ω. When we speak later of local coordinates, we mean that we take some element belonging to this family.

6. Rough upper bounds

To prove our results, we need some rough preliminary estimates (with some control of the remainder) of the ground state energy.

PROPOSITION 6.1. – Let Ωbe a bounded open set of R³ with smooth boundary ∂Ω. Let P_A,Ω^h,N be the Neumann operator onL²(Ω)associated to the Schrödinger operator with constant magnetic field (hD−A)² and let us assume that the vector magnetic field H = curl(A) is constant of intensityb.

Ifλ(h) = inf Sp(P_A,Ω^h,N)is the first eigenvalue ofP_A,Ω^h,N, then there exists a constantC0 such that

λ(h)bΘ0h+C0h^4/3. (6.1)

Remark 6.2. – It is not necessary to assume that the magnetic field is constant as the proof of this proposition will show. If one choose a pointx0such thatH(x0)is tangent to∂Ω, one can get the same result with b=|H(x0)|. One can then optimize by choosing a point of this kind such that|H(x0)|is minimal. IfΓHis the set introduced in (1.4) we will prove:

λ(h)Θ0 inf

x∈ΓH

H(x)h+C0h^4/3. (6.2)

This kind of estimate, with a more explicit, but not optimalC0, appears already in the appendix of [23].

Proof of Proposition 6.1. – As the flux of the magnetic field through the boundary is zero,

∂Ω

curl A(x)

|N(x) ds=

Ω

div curl(A)dx= 0, (6.3)

ifN is the interior normal unit of∂Ω, there existsx0∈∂Ωsuch that the vector magnetic field is tangent to∂Ωatx0:

curl A(x0)

|N(x0)

= 0.

We take local coordinates(y1, y2)in a neighbourhood ofW0ofx0in∂Ωsuch that the metric is δijatx0and_∂y^∂

1 is parallel to the vector magnetic field.

We consider the adapted coordinates y = (y1, y2, y3), with y3(x) =d(x, ∂Ω), and then gij(x0) =δij. IfAis the magnetic potential in the new coordinates,A·dx=A·dy, then

∂A3

∂y2

(x0)−∂A2

∂y3

(x0) =b,

∂A1

∂y3

(x0)−∂A3

∂y1

(x0) = 0, (6.4)

∂A2

∂y1

(x0)−∂A1

∂y2

(x0)) = 0.

We can find a gauge transformexpi^ϕ_h, withϕreal, such that, in (8.27), A1(x0) = 0, A2(x0) = 0, A3= 0.

We first estimate the error occuring when eliminating the terms vanishing to order3.

Ifu∈H¹(Ω)is such that

supp(u)⊂ x; |x−x0|h^δ (6.5)

withδ >0, and ifhis small enough, then, for someC >0, q_A^h(e^iϕhu)(1 +h^δC)q^h_A0(u) +C

h^3δ

q_A^h0(u)1/2

· u+h^6δu² , (6.6)

where

A⁰₁=R1(y), A⁰₂=−by3+R2(y), A⁰₃= 0, theRj(y)are polynomial functions, homogeneous of order two,

Rj(y) =

|α|=2

aj,αy^α,

and

q^h_A0(u) =

|y|Ch^δ, y₃>0

(hDyu−A⁰(y)u²dy.

(6.7)

We take the functionuin the form

u(y1, y2, y3) :=e^{−ib1/2y2ξ0/h1/2}v(y) with

v(y1, y2, y3) =h^−14−δϕ0(b^1/2h^−1/2y3)χ(4h^−δy3)χ

4h^−δ(y1²+y2²)^1/2 . (6.8)

Hereχis a cut off function equal to one on[−¹₂,¹₂]and supported in[−1,1],ϕ0is introduced in Section 2.2 andδ∈]0,¹₂[.

Observing that:

R+

Dtϕ0(t)²+(t−ξ0)ϕ0(t)²

dt= Θ0

R+

ϕ0(t)²dt,

and thatt→ϕ0(t)is an exponentially decreasing function at infinity, we get whenδ=¹₃, q^hAˆ⁰(u)(bΘ0h+Ch^4/3)u²,

(6.9) where

Aˆ⁰1= 0, Aˆ⁰2=−by3, Aˆ⁰3= 0.

We then have to compareq^h_Aˆ0(u)andq^h_A0(u). The first try could be to use:

q^h_A0(e^iϕhu)(1 +h^δC)q^h_Aˆ0(u) +C h^2δ

q^h_Aˆ0(u)1/2

· u+h^4δu² . (6.10)

This leads to error terms of sizeh^1+δ,h^12+2δ,h^4δandh^2−2δ. But this leads only, using (6.9) and (6.10) withδ=¹₃, to (6.1) withO(h^7/6)instead ofO(h^4/3)as expected.

In order to get effectively (6.1), we need to use (2.8). We have indeed to analyze more carefully the term which was bounded from above in (6.10) byh^2δ(q^h_Aˆ0(u))^1/2· u.

The terms which were estimated byh^12+2δ are of the form h^−12−2δ

yjyk

ξ0−b^1/2y3

h^1/2

ϕ²₀(b^1/2h^−1/2y3)χ²

4h^−δ(y²₁+y²₂)^1/2 dy, withjandkequal to1or2. But they actually vanish due to (2.8).

The other terms have actually better upper bounds. For example:

h^−12−2δ

y2y3

ξ0−b^1/2y3

h^1/2

ϕ²₀(b^1/2h^−1/2y3)χ(4h^−δy3)χ²

4h^−δ(y₁²+y₂²)^1/2 dy, can be estimated byO(h^1+δ). This achieves the proof of Proposition 6.1. ✷

This suggests strongly that the next term in the expansion ofλ(h)isO(h^4/3), but to go further we need to analyze the model more carefully in the neighborhood of the points of the boundary whereH(x)|N(x)vanishes. For this we need to enter more deeply into the geometry of the boundary in connection with the magnetic vector field, and this will be done in Section 8.

7. Rough lower bounds

We assume that the magnetic field is constant. Letu^h be an eigenfunction associated to the eigenvalueλ(h) = inf Sp(P_A,Ω^h,N).

7.1. A priori estimates

As proved in [12], we have, without the assumption (1.5), the following behavior ofu^h.

PROPOSITION 7.1. – Ift=t(x) =d(x, ∂Ω), then, for anyk∈N, there exists a constantCk

depending only onk, such that

t^k/2u^hCkh^k/4u^h (7.1)

and t^k/2(hD−A)u^hCkh^(k+2)/4u^h.

(7.2)

Proof. – Proposition 6.1 and the fact thatΘ0<1give the existence of constantsh0>0 and C0>1such that

C₀⁻¹bhbh−λ(h)C0bh, ∀h∈]0, h0].

(7.3)

Of course this is a very rough estimate.

Let us remark that (7.1) and (7.2) are valid whenk= 0. As for the2-dimensional case in [11], we proceed by recursion.

By changingtaway from the boundary, we can assume thatx→t(x) =d(x, ∂Ω)is extended as a C¹(Ω) function. By choosing suitable coordinates and after a gauge transform, we can assume that

A1(x) = 0, A2(x) =−b

2x3, A3(x) =b 2x2. Ast/∂Ω = 0, we have by integrating by parts and fork >0,

ihb

Ω

t^k|u^h|²dx=

Ω

t^k

(hD2−A2),(hD3−A3)

u^h·u¯^hdx

=

Ω

t^k (hD3−A3)u^h(hD2−A2)u^h−(hD2−A2)u^h(hD3−A3)u^h dx

+ihk

Ω

t^k−1 ∂t

∂x2

(hD3−A3)u^h− ∂t

∂x3

(hD2−A2)u^h

¯ u^hdx (7.4)

and

Ω

t^k(hD−A)u^h2dx=

Ω

λ(h)t^k|u^h|²−ihkt^k−1

(∇xt)·(hD−A)u^h

¯ u^h

dx.

(7.5)

Ifk= 1we get from (7.4) hb

Ω

t|u^h|²dx

Ω

t(hD−A)u^h2dx+Ch(hD−A)u^h· u^h, (7.6)

and then, we use (7.5) to get hb

Ω

t|u^h|²dxλ(h)

Ω

t|u^h|²dx+Ch(hD−A)u^h· u^h. (7.7)

So

hb−λ(h)

Ω

t|u^h|²dxCh

λ(h)u^h2. (7.8)

We obtain (7.1) with k= 1 from (7.3) and (7.8), and then it is easy to get (7.2) for k= 1 from (7.5) by using (7.3), (7.1) withk= 1and (7.2) withk= 0.

Ifk2,we get in the same way hb−λ(h)

Ω

t^k|u^h|²dxhkCt^−1+k2(hD−A)u^h· t^k/2u^h, (7.9)

which gives, using (7.3),

t^k/2u^hCkt^−1+k2(hD−A)u^h. (7.10)

Using (7.10) and (7.5) we get

t^k|(hD−A)u^h2Ckh

t^k/2u^h2+t^−1+k2|(hD−A)u^h2 . (7.11)

Then we can proceed by recursion. (7.2) fork=j−2and (7.10) fork=jgive (7.1) fork=j.

Formulas (7.11) withk=jand (7.2) fork=j−2give (7.2) withk=j. ✷ 7.2. A partition of unity

Let(χγ(z))γ∈Z³be a partition of unity ofR³.For example we can take χγ∈C^∞(R³;R) and supp(χγ)⊂γ+ [−1,1]³, ∀γ∈Z³,

γ

χ²_γ(z) = 1 and

γ

χγ(z)²<∞. (7.12)

Ifτ(h)is a function ofhsuch thatτ(h)∈]0, ε(Ω)[, whereε(Ω)is the geometric constant which is the maximalεfor the property that{d(x, ∂Ω)< ε}is a regular tubular neighborhood inΩ of∂Ω, we will define the functions

χγ,τ(h)(z) =χγ

z/τ(h)

, ∀γ∈Z³. (7.13)

So we get a new partition of unity such that

γ

χ²_γ,τ(h)(z) = 1,

γ

∇χγ,τ(h)(z)²Cτ(h)⁻², (7.14)

and

supp(χγ,τ(h))⊂τ(h)γ+

−τ(h), τ(h)3

. Then, for anyu∈H¹(Ω), we have:

q^h_A(u) =

γ

q_A^h(χγ,τ(h)u)−h²|∇χγ,τ(h)|u² . (7.15)

Let us define